Minimal matchings of point processes

Abstract

Suppose that red and blue points form independent homogeneous Poisson processes of equal intensity in $${{\mathbb {R}}}^d$$ R d . For a positive (respectively, negative) parameter $$\gamma $$ γ we consider red-blue matchings that locally minimize (respectively, maximize) the sum of $$\gamma $$ γ th powers of the edge lengths, subject to locally minimizing the number of unmatched points. The parameter can be viewed as a measure of fairness. The limit $$\gamma \rightarrow -\infty $$ γ → - ∞ is equivalent to Gale-Shapley stable matching. We also consider limits as $$\gamma $$ γ approaches 0, $$1-$$ 1 - , $$1+$$ 1 + and $$\infty $$ ∞ . We focus on dimension $$d=1$$ d = 1 . We prove that almost surely no such matching has unmatched points. (This question is open for higher d). For each $$\gamma <1$$ γ < 1 we establish that there is almost surely a unique such matching, and that it can be expressed as a finitary factor of the points. Moreover, its typical edge length has finite rth moment if and only if $$r<1/2$$ r < 1 / 2 . In contrast, for $$\gamma =1$$ γ = 1 there are uncountably many matchings, while for $$\gamma >1$$ γ > 1 there are countably many, but it is impossible to choose one in a translation-invariant way. We obtain existence results in higher dimensions (covering many but not all cases). We address analogous questions for one-colour matchings also.

There is no stationary cyclically monotone Poisson matching in 2d

“…Again in [9], it was observed by Holroyd, that by the triangle inequality the 1-local optimality condition (1.1) implies planarity. Hence, it is natural to consider the following modification of Question 1.1, which has been proposed in [10].…”

“…In [10,Theorem 7] it has been shown that stationary p-locally optimal matchings exist for p < 1. Our result, together with the one obtained in [13], complements the one of [10] in the regime p > 1 and d = 2, leaving unsolved the case p = 1 and of course the question on planarity.…”